P=W Phenomena

Harder, Andrew; Katzarkov, Ludmil; Przyjalkowski, Victor

doi:10.48550/arxiv.1905.08706

Cited by 4 publications

(4 citation statements)

References 16 publications

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“…Such a compactification exists for higher rank Tyurin degenerations at least so long as each of the components themselves have nef anti-canonical bundle. We believe that the restriction to this case is a necessary first step to understanding the "Mirror P " W conjecture" of [2] through the Gross-Siebert construction. The compactifications have the property that there are no broken lines passing in from infinity.…”

Section: Formation Of Landau-ginzburg Modelsmentioning

confidence: 96%

Towards the Doran-Harder-Thompson conjecture via the Gross-Siebert program

Barrott¹,

Doran²

2021

Preprint

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The Doran-Harder-Thompson "gluing/splitting" conjecture unifies mirror symmetry conjectures for Calabi-Yau and Fano varieties, relating fibration structures on Calabi-Yau varieties to the existence of certain types of degenerations on their mirrors. This was studied for the case of Calabi-Yau complete intersections in toric varieties in [12] for the Hori-Vafa mirror construction. In this paper we prove one direction of the conjecture using a modified version of the Gross-Siebert program. This involves a careful study of the implications within tropical geometry and applying modern deformation theory for singular Calabi-Yau varieties.

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Section: Formation Of Landau-ginzburg Modelsmentioning

confidence: 96%

Towards the Doran-Harder-Thompson conjecture via the Gross-Siebert program

Barrott¹,

Doran²

2021

Preprint

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“…The conjecture holds for g ≥ 2 and G = GL2, SL2 and PGL2 by [13], and for g = 2 and G = GLn, SLp with p prime by [16,17]. An enumerative approach has been proposed in [9], and other P=W phenomena have been studied in [61,60,72,67,68,55,36,35,37]. However, P=W phenomena for the original moduli spaces MB(X, G) and M Dol (X, G) have not been explored yet.…”

Section: Introductionmentioning

confidence: 99%

P=W conjectures for character varieties with symplectic resolution

Felisetti¹,

Mauri²

2020

Preprint

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We establish P=W and PI=WI conjectures for character varieties with structural group GLn and SLn which admit a symplectic resolution, i.e. for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W conjecture for resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-étale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, or the intersection theory of some singular Lagrangian cycles. In particular, we study in detail a Dolbeault moduli space which is specialization of the singular irreducible holomorphic symplectic variety of type O'Grady 6. PreliminariesIn this section we introduces preliminary notions and results which will be useful throughout the paper. For further details we refer to [3,29,30,18]. Perverse sheavesAn algebraic variety X is an irreducible separated scheme of finite type over C. Denote by D b c (X) the bounded derived category of Q-constructible complexes on X. Let D : D b c (X) → D b c (X) be the Verdier duality functor. The full subcategoriesX) of the t-structure is the abelian category of perverse sheaves. The truncation functors are denoted p τ ≤k : D b c (X) → p D b ≤k (X), p τ ≥k : D b c (X) → p D b ≥k (X), and the perverse cohomology functors are p H k := p τ ≤k p τ ≥k : D b c (X) → Perv(X).

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“…More generally, one may ask about the fundamental group of the complement of a divisor in a projective variety. Examples of importance in mirror symmetry are log Calabi-Yau varieties [7,8,10], which are quasi-projective varieties that are the complement of an anti-canonical divisor in a smooth projective variety. We consider this case when the ambient projective variety is a flag variety.…”

Section: Introductionmentioning

confidence: 99%

On the fundamental group of open Richardson varieties

Li,

Sottile,

Zhang

2019

Preprint

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We compute the fundamental group of an open Richardson variety in the manifold of complete flags that corresponds to a partial flag manifold. Rietsch showed that these log Calabi-Yau varieties underlie a Landau-Ginzburg mirror for the Langlands dual partial flag manifold, and our computation verifies a prediction of Hori for this mirror. It is log Calabi-Yau as it isomorphic to the complement of the Knutson-Lam-Speyer anti-canonical divisor for the partial flag manifold. We also determine explicit defining equations for this divisor.

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P=W Phenomena

Cited by 4 publications

References 16 publications

Towards the Doran-Harder-Thompson conjecture via the Gross-Siebert program

Towards the Doran-Harder-Thompson conjecture via the Gross-Siebert program

P=W conjectures for character varieties with symplectic resolution

On the fundamental group of open Richardson varieties

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